What do you even do in this quest? It is telling me to destroy the 4 tunnel entrances but the game doesn't give you even the most basic information on how to achieve this.

You need to make some Grapeshot bombs, talk to Cedric to get diagrams/buy bombs from him. Then when you approach the nest it will say something along the lines of "detonate nest" - detonate 4 nests and you'll be done.

Umm, you could always buy a book about Nekkers from the dwarf bookseller............

Note that there are actually 5 nests.
To save you some searching: They are all within the wood, none within caves.

Ah, okay, so they aren't in those tunnels right outside town where you encounter all the nekkers? That is what has been confusing me. I can't find anything in those tunnels and the quest doesn't have any map marker pointing out where you need to go.

Yes, the nests are in the woods. Use your Z key and they will light up.

Be careful! Those red spider-things are close by too!

Where does it do that? The quest doesn't mention bombs at all. The book mentions how to destroy the entrances.

The main problem here is that for some mysterious reason, the knowledge you gained is not added to your journal, but to your character sheet. That's a somewhat bizarre design decision.

Perhaps some hyperlinking between Nekker-related quests, Nekker info in the journal, and Nekker knowledge from books would have been a good idea. It's a bit spread out right now. Right outside town where you encounter nekkers sounds about right. You're in the right place when there's a nest that lights up on 'z', and you get the "destroy nest" option.

Why not just read the guide that came with the game? Other than the translation error of "Beehive bombs" instead of "Grapeshot bombs" - it explains it very well.

And for more detail, they're small mounds (as in not quite knee-high or so, we're not talking giant termite mounds) with a single opening on the top, surrounded by scattered bones, bloodstains, and burrowed Nekkers. Sheathe your sword, click the LMB when prompted to destroy the nest, then back off a bit so you don't get hurt by the blast. The mound will still be highlighted by the medallion even after you've destroyed the entrance.

If you find yourself ambushed by a dozen Nekkers all emerging from the ground at once, there's a pretty good chance you're at a mound. It's not a bad idea to have an appropriate bomb or two readied.

Found in the marshy parts of the wood. Mist along the woodland floor etc.

Maybe this will help. I hope so....

http://www.pubarticles.com/article-simple-ways-to-find-the-last-nekker-nest-in-the-witcher-2-1306461602.html

btw OP, you will need at least level 2 knowledge of nekkers to be able to blow up the tunnels. This knowledge can be gained by killing enough nekkers or by reading a book bought from the dwarf in Flotsam (I think the book can also be found if you explore within the temple in the Prologue)